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KVPY2018MathematicsApplication of Derivatives

The number of polynomials p : R → R satisfying p 0 = 0 , p x > x 2 for all x ≠ 0 and p ' ' 0 = 1 2 is

Options

  1. A0
  2. B1
  3. Cmore than 1 , but finite
  4. Dinfinite

Correct answer

A. 0

Step-by-step solution

We have, p x > x 2 , p 0 = 0 , p ' ' 0 = 1 2 Let g x = p x - x 2 g x > 0 , ∀ x ≠ 0 and g 0 = p 0 - 0 = 0 ⇒ x = 0 should be minima. ∵ g ' ' x should b e ≥ 0 at x = 0 Now, g ' x = p ' x - 2 x g ' ' x 2 = p ' ' x - 2 g ' ' 0 = p ' ' 0 - 2 = 1 2 - 2 = - 3 2 But g ' ' 0 ≥ 0 ∵ No polynomial exists.

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